Quantitative correlation inequalities via extremal power series

Many correlation inequalities for high-dimensional functions in the literature, such as the Harris–Kleitman inequality, the Fortuin–Kasteleyn–Ginibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. Previous work has used Markov semigroup arguments to obtain quantitative extensions of some of these correlation inequalities. In this work, we augment this approach with a new extremal bound on power series, proved using tools from complex analysis, to obtain a range of new and near-optimal quantitative correlation inequalities. These new results include: A quantitative version of Royen’s celebrated Gaussian Correlation Inequality (Royen, 2014). In (Royen, 2014) Royen confirmed a conjecture, open for 40 years, stating that any two symmetric convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, conceptually similar to Talagrand’s quantitative correlation bound for monotone Boolean functions over {0,1}^n (Talagrand in Combinatorica 16(2):243–258, 1996). We show that our quantitative version of Royen’s theorem is within a logarithmic factor of being optimal. A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space. This is a broad generalization of Talagrand’s quantitative correlation bound for functions from {0,1}^n to {0,1} under the uniform distribution (Talagrand in Combinatorica 16(2):243–258, 1996). In the special case of p -biased distributions over {0,1}^n that was considered by Keller, our new bound essentially saves a factor of p log (1/p) over the quantitative bounds given in Keller (Eur J Comb 33:1943–1957, 2012; Improved FKG inequality for product measures on the discrete cube, 2008; Influences of variables on Boolean functions. PhD thesis, Hebrew University of Jerusalem, 2009).